1 Introduction In the paper [2] S. Fomin and the first author have introduced a model for the cohomology ring of flag varieties of type A

ON SOME NONCOMMUTATIVE ALGEBRAS RELATED WITH K-THEORY OF FLAG VARIETIES, I

anatol n. kirillov and toshiaki maeno

Abstract

For any Lie algebra of classical type or type G₂ we define a K-theoretic analog of Dunkl’s elements, the so-called truncatedRuijsenaars-Schneider-Macdonald elements, RSM-elements for short, in the corresponding Yang-Baxter group, which form a com- muting family of elements in the latter. For the root systems of typeA we prove that the subalgebra of thebracket algebra generated by the RSM-elements is isomorphic to the Grothendieck ring of the flag variety. In general, we prove that the subalgebra gen- erated by the images of the RSM-elements in the corresponding Nichols-Woronowicz algebra is canonically isomorphic to the Grothendieck ring of the corresponding flag varieties of classical type or of type G₂. In other words, we construct the “Nichols- Woronowicz algebra model” for the Grothendieck Calculus on Weyl groups of classical type or typeG₂,providing a partial generalization of some recent results by Y. Bazlov.

We also give a conjectural description (theorem for typeA) of a commutative subalge- bra generated by the truncated RSM-elements in the bracket algebra for the classical root systems. Our results provide a proof and generalizations of recent conjecture and result by C. Lenart and A. Yong for the root system of typeA.

1 Introduction

In the paper [2] S. Fomin and the first author have introduced a model for the cohomology ring of flag varieties of type Aas a commutative subalgebra generated by the so-called trun- cated Dunkl elements in a certain (noncommutative) quadratic algebra. This construction has been generalized to other root systems in [5]. The main purpose of the present paper is to construct a K-theoretic analog of these constructions. More specifically, we introduce certain families of pairwise commuting elements in the Yang-Baxter group GK(B_n) or in the bracket algebra BE(B_n), which conjecturally generate commutative subalgebras in the bracket algebra BE(B_n) isomorphic to the Grothendieck ring of the flag varieties of type Bn. The corresponding results/conjectures for the flag varieties of other classical type root systems can be obtained from those for the typeB after certain specializations. There exists the natural surjective homomorphism¹ from the algebraBE(B_n) to the Nichols-Woronowicz

1It is believed that for a simply–laced (finite) Coxeter system (W, S) the corresponding bracket algebra BE(W, S) and the Nichols–Woronowicz algebraBW,Sare isomorphic as braided Hopf algebras. However, this is not the case in a non simply–laced case. For example, ifn≥3,the natural epimorphismBE(Bn)−→ BBn

has a non-trivial kernel in degree 6. In fact,Hilb(BE(Bn), t)−Hilb(BBn, t) = 4t⁶+· · ·.

algebra B_Bn of type B. One of our main results of the present paper states that the im- age of our construction in the Nichols-Woronowicz algebra B_Bn is indeed isomorphic to the Grothendieck ring of the flag variety of typeB_n.We also present a similar construction for the root system of typeG₂.These results can be viewed as a multiplicative analog/generalization for classical root systems and for G₂ of the “Nichols-Woronowicz algebra model” for the co- homology ring of flag varieties which has been constructed recently by Y. Bazlov [1].

In a few words the main idea behind the constructions of the paper can be described as follows. As it was mentioned, in [2] for type A and in [5] for other root systems, a re- alization of the (small quantum) cohomology ring of flag varieties has been invented. More specifically, the papers mentioned above present a model for the cohomology ring of flag varieties as a commutative subalgebra generated by the so-called Dunkl elements in a certain (noncommutative) algebra. The main ingredient of this construction is based on some very special solutions to the classical Yang-Baxter equation (for type A) and classical reflection equations (for typesB, C andG2). Our original motivation was to study the related algebras and groups which correspond to the “quantization” of the solutions to the classical Yang- Baxter type equations mentioned above, in connection with classical and quantum Schubert and Grothendieck Calculi. In more detail, we define the group of “local set-theoretical so- lutions” to the quantum Yang-Baxter equations of type B_n or of type G₂, together with the distinguished set of pairwise commuting elements in the former, the so-called truncated Ruijsenaars-Schneider-Macdonald elements. The latter is a relativistic or multiplicative gen- eralization of the Dunkl elements. For applications to theK-theory, we specialize the general construction to the bracket algebraBE(B_n) and algebra BE(G₂).

Summarizing, the main construction of our paper presents a conjectural description of the Grothendieck ring K(G/B) corresponding to flag varieties G/B of classical types (or G₂-type) to be a commutative subalgebra in the corresponding bracket algebra generated by the truncated RSM-elements. To be more specific, we construct in the algebra BE(Bn) a pairwise commuting family of elements, multiplicative or relativistic Dunkl elements, and state a conjecture about the complete list of relations among the latter.

Using some properties of the Chern homomorphism, we prove our conjecture for the root systems of type A. To our best knowledge, for the root systems of type A a similar description of the Grothendieck ring was given by C. Lenart and A. Yong [9], [10], however without reference to the Yang-Baxter theory.

The main problem to prove relations between the RSM-elements in the bracket algebra BE(B_n) appears to be show that the intersection of kernels of all the “braided derivations”

∆ij,∆_ij 1 ≤ i < j ≤ n, and ∆i, 1 ≤ i ≤ n, acting on the algebra BE(Bn) contains only constants, see Section 5. At this point we pass to the Nichols-Woronowicz algebraB_Bn where the corresponding property of the braided derivations is guaranteed, [1]. Since as mentioned above, there exists the natural epimorphism of braided Hopf algebras BBn −→ BE(Bn), to check the corresponding relations in the Nichols-Woronowicz algebra B_Bn seems to be a good step to confirm our conjectures. To prove the needed relations in the algebra B_Bn, we develop a multiplicative analog/generalization of the Nichols-Woronowicz algebra model for cohomology ring of flag varieties recently introduced by Y. Bazlov [1].

Let us describe briefly the content of our paper.

Section 2 is devoted to a general construction of commuting family of elements in the group GK(B) generated by local set-theoretical solutions to the family of quantum Yang- Baxter equations of typeB, see Definition 2.1 for precise formulation. This construction lies at the heart of our approach. In the case of type A (i.e. if g_ij =h_i = 1 for all i and j) and the Calogero-Moser representation (i.e. h_ij = 1 +∂_ij) of the bracket algebra BE(A_n−1), the elements Θ^A₁ⁿ⁻¹,· · ·,Θ^A_nⁿ⁻¹ correspond to the (rational) truncated (i.e. without differential part) Ruijsenaars-Schneider-Macdonald operators. It seems an interesting problem to classify all irreducible finite dimensional representations of the groups GK(X), (X = An−1, Bn, ...) together with a simultaneous diagonalization of the operators Θ^X₁ ,· · ·,Θ^X_n in these represen- tations.

In Section 3 we apply the result of Section 2 (Key Lemma) to construct some distinguished multiplicative analogue Θ_j^An−1 := Θ^A_jⁿ⁻¹(x) of the Dunkl elements θ_j^An−1, 1 ≤ j ≤ n, in the bracket algebra BE(A_n−1). It happened that our elements Θ^A_jⁿ⁻¹ coincide with the K- theoretic Dunkl elements 1−κ_j introduced by C. Lenart and A. Yong in [9]. Proof of the statement that the elementsκ₁,· · ·, κ_n form a family of pairwise commuting elements in the algebraBE(An−1) given in [9], appears to be quite long and involved. On the other hand, our

“Yang-Baxter approach” enables us to give a simple and transparent proof that the elements Θ^A_jⁿ⁻¹ mutually commute, as well as to describe relations among these commuting elements in the algebra BE(An−1). On this way we come to the main result of Section 3, namely Theorem A The subalgebra in BE(An−1) generated by the elements Θ_j^An−1, 1 ≤ j ≤ n, is isomorphic to the Grothendieck ring of the flag varieties of type A.

In particular,

Theorem B The following identity in the algebra BE(An−1) holds:

Xn

j=1

(Θ^A_jⁿ⁻¹(x))^k =n for any k ∈Z.

Theorem A was stated as Conjecture 3.4 in [9]. We also state a positivity conjecture as Conjecture 3.13, which relates the elements Θ^A_jⁿ⁻¹ to the Grothendieck Calculus on the group GL_n.This conjecture is a restatement of non-negativity conjectures from [2], Conjecture 8.1, and [9], Conjecture 3.2., in our setting.

It should be emphasized that there are a lot of possibilities to construct a mutually commuting family of elements in the algebra BE(A_n−1) which generate a subalgebra iso- morphic to the Grothendieck ring K(Fl_n). For example one can take the elements E₁ :=

exp(θ₁^An−1),· · ·, E_n := exp(θ_n^An−1). It is easy to see that E_j 6= Θ_j for all j, however connec- tions between Grothendieck polynomials and the elements E₁,· · ·, E_n are not clear for the authors.

Our method to describe the relations between the elements Θ^A_jⁿ⁻¹ is based on the study of the Chern homomorphism which relates the K-theory to the cohomology theory of flag

varieties, and moreover, on description of thecommutative quotientof the algebraBE(A_n−1), see Subsection 3.2.

In Section 4 we study the B_n-case. First of all we introduce a modified version BE(B_n) of the algebra BE(B_n), which was introduced in our paper [5]. Namely, we add additional relations in degree four, see Definition 4.1, (6).In fact we have no need to use these relations in order to describe relations between Dunkl elements θ^B₁ⁿ,· · ·, θ_n^Bn in the algebra BE(B_n).

However, to ensure that theB₂ Yang-Baxter relations h_ij h_i g_ij h_j =h_j g_ij h_i h_ij are indeed satisfied, the relations (6) are necessary. Another reason to add relations (6) is that these relations are satisfied in the Nichols-Woronowicz algebra B_Bn. However, we would like to repeat again that if n≥3, then the natural homomorphism of algebras BE(B_n)→ B_Bn has a non-trivial kernel.

The main results of Section 4 are:

(1) construction of a multiplicative analogue Θ^B_jⁿ of the B_n-Dunkl elements θ_j^Bn, see Definition 4.4;

(2) proof of the fact that the RSM-elements Θ^B_jⁿ,1≤j ≤n, form a pairwise commuting family of elements in the algebraBE(B_n).

Finally we give a conjectural description of all relations between the elements Θ^B_jⁿ. Here we state this conjecture in the following form.

ConjectureThe following identity in the algebra BE(B_n) holds:

Xn

j=1

(Θ_j(x, y)^Bn+ (Θ_j(x, y)^Bn)⁻¹)^k =n 2^k for all k ∈Z_≥0.

In Section 5 we discuss on a model for the Grothendieck ring of flag varieties in terms of the Nichols-Woronowicz algebra for the classical root systems. Our construction is a K- theoretic analogue of Bazlov’s result [1]. Our second main result proved in Section 5 is:

Theorem C Let ϕ:BE(B_n)→ B_Bn be a natural homomorphism of algebras. Then ϕ(F(Θ^B₁ⁿ(x, y),· · ·,Θ^B_nⁿ(x, y))) = 0

for any Laurent polynomial F from the defining ideal of the Grothendieck ring of the flag variety of type B_n.

Theorem C implies the corresponding results for other classical root systems after some specializations. The Nichols-Woronowicz algebra B_X treated in this paper is a quotient of the algebra GK(X) for the classical root system X. In particular, the result for A_n−1 is a consequence of Theorem A, but the argument in Section 5 is another approach based on the property of the Nichols-Woronowicz algebra, which works well for the root systems other than of type A_n−1.The idea of the proof is to construct the operators on the Nichols-Woronowicz algebra which induce isobaric divided difference operators on the commutative subalgebra generated by the RSM-elements.

The main interest of this paper is concentrated on the classical root systems, for which we can use advantages of explicit handling, particularly in order to construct the RSM-elements.

Though most of the ideas in this paper are expected to be applicable to an arbitrary root system, to develop the general framework including the exceptional root systems is a matter of concern for the forthcoming work. However, the simplest exceptional root systemG₂ can be dealt with in similar manner to the case of the classical root systems. In the last section, we formulate the Yang-Baxter relations and define the RSM-elements for the root system of typeG₂. The argument in Section 5 again works well, so the Nichols-Woronowicz model for the Grothendieck ring of the flag variety of typeG₂ is presented.

2 Key Lemma

Definition 2.1 Let GK(B_n) be a group generated by the elements {h_ij, g_ij | 1≤i < j ≤n}

and {h_i |1≤i≤n}, subject to the following set of relations:

• h_ij h_kl=h_kl h_ij, g_ij g_kl =g_kl g_ij, h_k h_ij =h_ij h_k, h_k g_ij =g_ij h_k, if all i, j, k, l are distinct;

hi hj =hj hi, if 1≤i, j ≤n; hij gij =gij hij, if 1≤i < j≤n;

• (Mixed Yang–Baxter relations)

(1) h_ij h_ik h_jk =h_jk h_ik h_ij, (2) h_ij g_ik g_jk =g_jk g_ik h_ij, (3) h_ik g_ij g_jk =g_jk g_ij h_ik, (4) hjk gij gik =gik gij hjk, if 1≤i < j < k≤n;

• (B₂ quantum Yang-Baxter relation)

hij hi gij hj =hj gij hi hij, if 1≤i < j≤n.

Definition 2.2 Define the following elements in the groupGK(B_n) : Θ_j = (

Y1

i=j−1

h⁻¹_ij ) h_j ( Yn

i=1,i6=j

g_ij) h_j (

j+1Y

k=n

h_jk), (2.1)

for 1≤j ≤n. In the RHS of (2.1) it is assumed that gij =gji. Theorem 2.3 (Key Lemma )

Θ_iΘ_j = Θ_jΘ_i for all 1≤i, j ≤n.

Proof. Induction plus a masterly use of the Yang-Baxter relations, see defining relations in the definition of the groupGK(Bn).See the proof of Corollary 3.3 and Example 2.5 (2) below.

Remark 2.4 It’s not difficult to see that Y

1≤j≤k

Θ_j = Yk

j=1

(h_j Yn

s=j+1

g_js

j−1Y

s=1

g_sj h_j) Yk

j=1

(

k+1Y

s=n

h_js).

In particular,

Yn

j=1

Θ_j = ( Yn

k=1

(Y

j≤k

g_jk) h_k)².

Example 2.5 (1) Take n = 2. Then Θ₁ = h₁ g₁₂ h₁ h₁₂ and Θ₂ = h⁻¹₁₂ h₂ g₁₂ h₂. Let us check that Θ₁ and Θ₂ commute. Indeed, using the B₂-quantum Yang-Baxter relation h12 h1 g12 h2 =h2 g12 h1 h12 and the commutativity relation h1 h2 =h2 h1, we see that

Θ₁ Θ₂ =h₁ g₁₂ h₁ h₂ g₁₂ h₂ =h⁻¹₁₂ (h₁₂ h₁ g₁₂ h₂)h₁ g₁₂ h₂

=h⁻¹₁₂ h₂ g₁₂ h₁(h₁₂ h₁ g₁₂ h₂) = h⁻¹₁₂ h₂ g₁₂ h₁ h₂ g₁₂ h₁ h₁₂ = Θ₂ Θ₁ = (h₁ g₁₂ h₂)². (2) Take n= 3. Then we have

Θ₁ =h₁ g₁₂ g₁₃ h₁ h₁₃ h₁₂, Θ₂ =h⁻¹₁₂ h₂ g₁₂ g₂₃ h₂ h₂₃, Θ₃ =h₂₃⁻¹ h⁻¹₁₃ h₃ g₁₃ g₂₃ h₃, and

Θ₁ Θ₂ Θ₃ = (h₁ g₁₂ h₂ g₁₃ g₂₃ h₃)².

Let us illustrate the main ideas behind the proof of Key Lemma by the following example.

Θ₁ Θ₃ Θ⁻¹₁ =h₁ g₁₂ g₁₃ h₁ h₁₃ h₁₂ h⁻¹₂₃ h⁻¹₁₃ h₃ g₁₃ g₂₃ h₃ h⁻¹₁₂ h₁₃⁻¹ h⁻¹₁ g⁻¹₁₃ g⁻¹₁₂ h⁻¹₁

=h₁ g₁₂ g₁₃ h₁ h⁻¹₂₃ h₃ g₂₃ g₁₃ h₃ h⁻¹₁₃ h⁻¹₁ g₁₃⁻¹ g₁₂⁻¹ h⁻¹₁ (by (1) and (2))

=h₁ h⁻¹₂₃ g₁₃ g₁₂ h₃ g₂₃ h⁻¹₁₃ h₃ g⁻¹₁₂ h⁻¹₁ (by (4) and B₂-YBE)

=h₁ h⁻¹₂₃ g₁₃ h₃ h⁻¹₁₃ g₂₃ h₃ h⁻¹₁ (by (3))

= Θ3 (by B2-YBE).

We define the groups GK(A_n−1) and GK(D_n) to be the quotients of that GK(B_n) by the normal subgroups generated respectively by the elements{h_i, g_ij,1≤i < j ≤n}and{h_i,1≤ i ≤ n}. The group GK(G₂) will be defined in Section 6. We expect that the subgroup in GK(B_n) generated by the elements Θ^B₁ⁿ,· · ·,Θ^B_nⁿ is isomorphic to the free abelian group of rankn.It seems an interesting problem to construct analogues of the group GK(B_n) and the elements Θ^B₁ⁿ,· · ·,Θ^B_nⁿ for any (finite) Coxeter group.

Question 2.6 Does there exist a finite-dimensional faithful representation of the group GK(X), X =A_n−1, B_n, ... ?

3 Algebras GK (A

) and BE(A

)

3.1 Definitions and main results

(i)Algebra GK(A_n−1)

Definition 3.1 Let R be a Q-algebra. Define the algebra GKR(An−1) as an associative algebra over R generated by the elements h_ij(x), 1 ≤ i 6= j ≤ n, x ∈ R, subject to the relations (0)−(4) :

(0) hij(x)hji(x) = 1,

(1) hij(x)hij(y) =hij(x+y); in particular, hij(x)hij(−x) = 1, (2) h_ij(x)h_kl(y) = h_kl(y)h_ij(x), if i, j, k, l are distinct,

(3) h_ij(x)h_jk(y) +h_ik(x+y) = h_jk(y)h_ik(x) +h_ik(y)h_ij(x), h_jk(y)h_ij(x) +h_ik(x+y) =h_ik(x)h_jk(y) +h_ij(x)h_ik(y), if 1≤i < j < k≤n,

(4) h_ik(x) (h_ij(x)−h_ik(y)) h_ij(y) = h_ij(y) (h_ij(x)−h_ik(y))h_ik(x), if 1≤i < j < k≤n.

For any element z ∈ R we denote by GK(An−1)[z] (resp. GK(An−1)) the algebra over Q generated by the elementsh_ij(z) and h_ij(−z), (resp. h_ij(1) andh_ij(−1)), 1≤i6=j ≤n.

Lemma 3.2 (Quantum Yang-Baxter equation) The following relations in the algebra GK(A_n−1)[z]

h_ab(z)h_ac(z)h_bc(z) = h_bc(z)h_ac(z)h_ab(z), 1≤a < b < c ≤n, (3.2) are a consequence of the relations (0)−(4) in the algebra GK(A_n−1)[z].

Corollary 3.3 Define elements Θ^A_jⁿ⁻¹(z), j = 1, . . . , n, in the algebra GK(A_n−1)[z] as fol- lows:

Θ^A_jⁿ⁻¹(z) =h⁻¹_j−1,j(z)· · ·h⁻¹_1j(z) h_jn(z)· · ·h_j,j+1(z), 1≤j ≤n. (3.3) Then

Θ^A_jⁿ⁻¹(z)Θ^A_kⁿ⁻¹(z) = Θ_k^An−1(z)Θ^A_jⁿ⁻¹(z), for all 1 ≤j, k ≤n.

Proof. It is enough to check that if 1≤i≤j ≤n, then Θ_i Θ_j Θ⁻¹_i = Θ_j. By definition,

Θ_i Θ_j Θ⁻¹_i =h⁻¹_i−1,i· · ·h⁻¹_1,i h_i,n· · ·h_i,i+1 h⁻¹_j−1,j· · ·h⁻¹_i+1,j h⁻¹_i,j h⁻¹_i−1,j· · ·h⁻¹_1,j h_j,n· · ·h_j,j+1 h⁻¹_i,i+1h_i,i+2⁻¹ · · ·h⁻¹_i,n h_1,i· · ·h_i−1,i.

Using local commutativity relations, see Definition 3.1 (2), we can move the factor h⁻¹_i,i+1 to

the left till we have touched on the factor h⁻¹_i,j . As a result, we will come up with the triple product:

h⁻¹_i+1,j h⁻¹_i,j h⁻¹_i,i+1,

which is equal, according to the Yang-Baxter relation (3.2),to the product h⁻¹_i,i+1 h⁻¹_i,j h⁻¹_i+1,j.

Now we can move the factor h⁻¹_i,i+1 to the left to cancel it with the term h_i,i+1, which comes from the rightmost factor in the element Θ_i.

As a result, we will have

Θ_i Θ_j Θ⁻¹_i =h⁻¹_i−1,i· · ·h_i,i+2 h⁻¹_j−1,j· · ·h⁻¹_i+2,j h⁻¹_i,j · · ·h⁻¹_j,j+1 h⁻¹_i,i+2· · ·h_i−1,i.

Now we can move to the left the factor h⁻¹_i,i+2 till we have touched on the factor h⁻¹_i,j to give the triple product

h⁻¹_i+2,j h⁻¹_i,j h⁻¹_i,i+2,

which is equal to h⁻¹_i,i+2 h⁻¹_i,j h⁻¹_i+2,j.Now we can move the factor h⁻¹_i,i+2 to the left to cancel it with the corresponding factor h_i,i+2, and so on.

It is readily seen that finally we will come to the element Θ_j. It is clear thatQ_n

j=1Θ^A_jⁿ⁻¹(z) = 1.

Remark 3.4 Let Θ_j(z) := Θ^A_jⁿ⁻¹(z). Then it is not true that Θ_j(x)Θ_k(y) = Θ_k(y)Θ_j(x), if j 6=k, x6=y.

Remark 3.5 Though the algebraGK(A_n−1)[z] can be constructed as a quotient of the group algebraQhGK(A_n−1)i, they are not isomorphic.

Theorem 3.6 (Main theorem, the case of algebra GK(A_n−1)[z]) Yn

j=1

(1 + (1−Θ^A_jⁿ⁻¹(z))t) = 1. Equivalently, Yn

j=1

(1 + Θ^A_jⁿ⁻¹(z)t) = (1 +t)ⁿ. (3.4)

This theorem is equivalent to:

Theorem 3.7 Let G^A_jⁿ⁻¹ = Θ^A_jⁿ⁻¹(z)−1, 1≤j ≤n. Then, after the substitution z = 1, e_j(G^A₁ⁿ⁻¹, . . . , G^A_nⁿ⁻¹) = 0, 1≤j ≤n

is the complete list of relations in the algebraGK(A_n+1)among the elementsG^A₁ⁿ⁻¹, . . . , G^A_nⁿ⁻¹. Here, e_j is the j-th elementary symmetric polynomial.

The proof is given in Subsection 3.2. It is based on the properties of the Chern homomor- phism.

Corollary 3.8 The algebra over Z generated by the elements G^A₁ⁿ⁻¹|_z=1, . . . , G^A_nⁿ⁻¹|_z=1, is canonically isomorphic to the integral Grothendieck ring K(Fl_n) of the flag manifold of type A_n−1.

(ii) Algebra BE(An−1)

Definition 3.9 ([2])Define algebraBE(A_n−1) (denoted byE_nin[2])as an associative algebra over Z with generators x_ij, 1≤i6=j ≤n, subject to the following relations

(0) x_ij +x_ji = 0, 1≤i6=j ≤n, (1) x²_ij = 0, 1≤i6=j ≤n,

(2) x_ij x_jk+x_jk x_ki+x_ki x_ij = 0, if all i, j, k are distinct.

The Dunkl elements θ_j, j = 1, . . . , n, in the algebra BE(A_n−1) are defined by θ_j :=

θ^A_jⁿ⁻¹ =P

i6=jxij.

The Dunkl elements form a pairwise commuting family of elements in the algebraBE(A_n−1), [2], and generate a commutative subalgebra in BE(A_n−1), which is canonically isomorphic to the cohomology ring H^∗(Fln) of the flag variety Fln of type An−1,[2].

For an element t of a Q-algebra R, define h_ij(t) = 1 +tx_ij = exp(tx_ij)∈BE(A_n−1)⊗R.

Lemma 3.10 The elements h_ij(t),1 ≤ i, j ≤ n, satisfy the all relations (0) −(4) of the definition of the algebra GKR(An−1).

We will use the same notation Θ^A_jⁿ⁻¹, 1 ≤ j ≤ n, to denote the elements in the algebra BE(An−1) defined by the formula (3.3). It follows from Corollary 3.3 that they form a pairwise commuting family of elements in the algebra BE(A_n−1).

It’s clear that Θ^A_jⁿ⁻¹(z) = 1 +z θ_j^An−1+· · ·, and the product in the RHS of (3.3) may be written as follows:

Θ^A_jⁿ⁻¹(z) =X

(−1)^sxb1,j xb2,j· · ·xbs,j xj,a1 xj,a2· · ·xj,ar z^r+s, (3.5) where the sum runs over the all sequences of integers (a₁ > a₂ > · · · > a_r) and (b₁ > b₂ >

· · ·> b_s) such that n≥a₁ > a_r > j > b₁ > b_s ≥1; cf. [9, Section 2].

Remember that G^A_jⁿ⁻¹ := Θ^A_jⁿ⁻¹ −1, 1≤j ≤n.

Definition 3.11 Letw∈S_nbe a permutation. Define the Grothendieck polynomialG_w(X_n)∈ Z [Xn] to be a unique polynomial of the form Gw(Xn) =P

α⊂δncα(w) x^α such that

G_w(G^A₁ⁿ⁻¹, . . . , G^A_nⁿ⁻¹)·id =w (3.6) in the Bruhat representation of the algebra BE(An−1) (see [2, Section 3.1] ), where δn :=

(n−1, n−2, . . . ,1,0) and X_n := (x₁, . . . , x_n).

It is not difficult to see that the Grothendieck polynomials defined here coincide with those introduced in [6], see also [10].

Corollary 3.12 Let u∈S_n and v ∈S_n be two permutations. Assume that in the group ring ZhS_ni of the symmetric group S_n we have the following equality:

G_u(G^A₁ⁿ⁻¹, . . . , G^A_nⁿ⁻¹)·v = X

w∈Sn

c^w_u,v w.

Then the coefficient c^w_u,v is equal to the multiplicity of the Grothendieck polynomial G_w(X_n) in the product of G_u(X_n) and G_v(X_n) :

Gu(Xn) Gv(Xn) = X

w∈Sn

c^w_u,v Gw(Xn) in the Grothendieck ring K(Fl_n) of the flag manifold of type A_n−1.

Conjecture 3.13 For any permutation w ∈ S_n the value of the Grothendieck polynomial Gw(x1, . . . , xn) after the substitutionx1 :=G₁^An−1, . . . , xn :=G^A_nⁿ⁻¹,andz = 1, can be written as a linear combination of monomials in x_ij’s , 1 ≤ i < j ≤ n, with non-negative integer coefficients.

Example 3.14 (Grothendieck-Pieri formula in the algebra BE(A_n−1), cf [10]) 1 +G_(k,k+1)(G₁, . . . , G_n) = Y

1≤j≤k

Θ_j = Yk

j=1 k+1Y

s=n

h_js =XY^r

j=1

x_aj,bj,

where the sum runs over all sequences of integers (1 ≤ a1 ≤ · · · ≤ ar ≤ k) and (b1, . . . , br) such thatk < b_j ≤n, j = 1, . . . , r, and a_i =a_i+1 ⇒b_i > b_i+1.

Example 3.15 Taken = 3, then

Θ₁ := Θ^A₁²(1) =h₁₃(1) h₁₂(1) = 1 +x₁₂+x₁₃+x₁₂ x₁₃,

Θ2 := Θ^A₂²(1) =h⁻¹₁₂(1) h23(1) = 1−x13+x23−x13 x12−x23 x13, Θ₃ := Θ^A₃²(1) =h⁻¹₂₃(1) h₁₃⁻¹(1) = 1−x₁₃−x₂₃+x₂₃ x₁₃.

As a preliminary step, we compute the elementary symmetric polynomials e_k(Θ₁,Θ₂,Θ₃), k = 1,2,3. Indeed, it’s easily seen from the formulae above that Θ1 + Θ2 + Θ3 = 3 and Θ₁ Θ₂ Θ₃ = 1.To computee₂(Θ₁,Θ₂,Θ₃),all one has to do is to apply the following relation

h₁₂ h⁻¹₂₃ =h⁻¹₂₃ h₁₃+h⁻¹₁₃ h₁₂−1,

where we put by definitionh_ij :=h_ij(1). The former equality follows from the relation (3) in Definition 3.1. Hence,

e₂(Θ₁,Θ₂,Θ₃) = h₁₃ h₂₃+h₁₃ h₁₂ h⁻¹₂₃ h⁻¹₁₃ +h⁻¹₁₂ h⁻¹₁₃

=h₁₃ h₂₃+h₁₃ h⁻¹₂₃ +h₁₂ h₁₃⁻¹−1 +h⁻¹₁₂ h⁻¹₁₃ = 2h₁₃+ 2h⁻¹₁₃ −1 = 3.

To continue, let us list the Grothendieck polynomialsG_w(x) corresponding to the symmetric group S₃:

Gid(x) = 1, Gs1(x) =x1, Gs2(x) = x1+x2 +x1x2,

G_s1s2(x) = x₁x₂, G_s2s1(x) = x²₁, G_w0(x) = x²₁x₂.

Now let us consider the substitution x_j = G_j = Θ_j(1) − 1, j = 1,2,3. More explicitly, G1 =x12+x13+x13 x12 and G2 =−x12+x23−x13 x12−x23 x13. Therefore,

G_s2(G₁, G₂) = x₁₃+x₂₃+x₁₃ x₂₃, G_s1s2(G₁, G₂) =x₁₃ x₂₃+x₂₃ x₁₃, G_s2s1(G₁, G₂) =x₁₂ x₁₃+x₁₃ x₁₂,

G_w0(G₁, G₂) =x₁₂ x₁₃ x₂₃+x₁₃ x₁₂ x₁₃+x₁₃ x₂₃ x₁₃+x₁₃ x₁₂ x₁₃ x₂₃.

Finally, let us consider the commutative subalgebra inBE(A₂)⊗Qgenerated by the elements Ej := exp(θj), j = 1,2,3. It’s not difficult to check that

2E₁ =h₁₃ h₁₂+h₁₂ h₁₃, 2E₂ =h⁻¹₁₂ h₂₃+h₂₃ h⁻¹₁₂, 2E₃ =h⁻¹₂₃ h⁻¹₁₃ +h⁻¹₁₃ h⁻¹₂₃.

It is an easy matter as well to see that the subalgebra inBE(A₂)⊗Q generated overQ by the elements E_i, i = 1,2,3, is isomorphic to the algebra Q[Θ₁,Θ₂,Θ₃]. In particular, for all symmetric polynomials f(x1, x2, x3) we have

f(1−E₁,1−E₂,1−E₃) = 0.

Proposition 3.16 The subalgebra in BE(An−1)⊗Qgenerated by the elementsEi := exp(θi), 1≤i≤n, is isomorphic to the algebra over Q generated by the elements Θ_j^An−1, 1≤j ≤n.

In particular, the complete list of relations among the elements 1−E1, . . . ,1−En in the quadratic algebra BE(A_n−1) is given by

ei(1−E1, . . . ,1−En) = 0,

fori= 1,· · ·, n.Thus the commutative subalgebra generated by the elementsexp(θ₁), . . . ,exp(θ_n) is isomorphic to the rational Grothendieck ring K(Fln)⊗Q of the flag manifold Fln of type A_n−1.

However, it seems that there are no direct connections of the elementsE_j’s with the Grothendieck Calculus.

Remark 3.17 More generally, letQ(t)6= 0 be a polynomial such that Q(0) = 0. Define the elements q_i := 1 +Q(θ_i), 1≤i≤n, in the algebra

BE(An−1). It’s clear that the elements q1, . . . , qn pairwise commute, and e_i(q₁−1, . . . , q_n−1) = 0, 1≤i≤n.

Remark 3.18 (Quantum Grothendieck Calculus)

It is easy to see that the relations in Definition 3.1 are still true, if we replace the condition (1) in Definition 3.9 by the following one

(1⁰) x²_ij =qij, 1≤i < j ≤ n, where the parameters qij are assumed to commute with all the generators x_kl, 1≤k < l≤n.

The algebra over Z[ q_ij | 1 ≤ i < j ≤ n] generated by the elements x_ij, 1 ≤ i 6= j ≤ n, subject to the relations (0), (1⁰) and (2), is called the quantized bracket algebra and denoted byqBE(A_n−1), cf. [2, Section 15] and [3].

As a corollary we see that the elements Θ^q_j, 1≤j ≤n, defined by the formula (3.2),form a pairwise commuting family of elements in the algebra qBE(An−1).

Problem 3.19 Describe the commutative subalgebras in the quantized algebra qGK(A_n−1) generated by

(1) Θ^q₁(1), . . . ,Θ^q_n(1),

(2) Ee₁ := exp(θ₁), . . . ,Ee_n := exp(θ_n).

3.2 Chern homomorphism

Denote byH :=BE(A_n−1)^ab⊗Qthe quotient of the algebraBE(A_n−1) by its commutant. It is known, [2, Proposition 4.2], that the algebraBE(A_n−1)^ab has dimensionn!, and its Hilbert polynomial is given by

Hilb(BE(A_n−1)^ab, t) = (1 +t)(1 + 2t)· · ·(1 + (n−1)t).

Denote by 1 +H⁺ the multiplicative monoid generated by the elements of the form 1 +h, where h∈ H does not have the term of degree zero.

Proposition 3.20 Let R⁽ⁿ⁻¹⁾ be the subspace of the commutative subalgebra

R=Q[θ₁, . . . , θ_n]⊂BE(A_n−1)⊗Q whose elements are of degree ≤n−1. Then the subspace R⁽ⁿ⁻¹⁾ is injectively mapped into H by the quotient homomorphism BE(An−1)⊗Q→ H.

Proof. Since the algebra R is isomorphic to the coinvariant algebra of the symmetric group, the monomials

θⁱ₁¹· · ·θⁱ_n−1ⁿ⁻¹, 0≤ik ≤n−k,

form a linear basis ofR. The linear mapR⁽ⁿ⁻¹⁾ → Hinduced by the quotient homomorphism is a homomorphism between S_n-modules. Hence, it is enough to show the images of the monomials θⁱ₁¹· · ·θ_n−1ⁱⁿ⁻¹ do not vanish in H for (i1, . . . , in−1) such that P_n−1

k=1ik = n−1 and i₁ ≥ i₂ ≥ · · · ≥ i_n−1. We expand the monomials θⁱ₁¹· · ·θ_n−1ⁱⁿ⁻¹ of this form in the algebra BE(An−1)⊗Qby using the Pieri formula proved by Postnikov [11], (first conjectured in [2]).

The Pieri formula shows that

e_k(θ₁, . . . , θ_m) =gX

[i₁j₁]· · ·[i_kj_k], wherePf

stands for the multiplicity-free sum, and (i₁, j₁), . . . ,(i_k, j_k) run over all pairs such that ia≤m < ja ≤n, a= 1, . . . , k, and allia’s are distinct.

On the other hand, the monomials of form

[i₁j₁]· · ·[i_kj_k], i_a < j_a (a= 1, . . . , k), j₁ < j₂ <· · ·< j_k,

give a linear basis ofH([3, Corollary 10.3]). By the involutionω: [ij]7→[n+ 1−j n+ 1−i], we have a linear basis of form

[i₁j₁]· · ·[i_kj_k], i_a < j_a (a= 1, . . . , k), i₁ < i₂ <· · ·< i_k. (3.7) For each monomial expression [i1j1]· · ·[ikjk] in H, we define

µ([i₁j₁]· · ·[i_kj_k]) :=

Xk

m=1

(j_m−i_m).